Cross Multiply

The common method of multiplying numbers in the form of a fraction is known as cross multiplication. Cross multiplication involves multiplying the numerators of one fraction by the denominator of the other across an equation. This method can be easily applied when solving linear equations with two variables. For example:

(a)1x + (b)1y + (c)1 = 0
(a)2x + (b)2y + (c)2 = 0

If cross-multiplication is performed correctly, we can quickly obtain the values of x and y by simplifying the equation.
 

Cross multiplication is primarily used with fractions, either to determine the greater fraction or to check if they are equal. This method can help solve equations when dealing with complex fractions. Given below are the different steps involved in this process:

For example: 5/8 = 10/16

Step 1: To begin, multiply the fraction's numerator on the right by the fraction's denominator on the left. Multiplying 10 × 8 = 80.

Step 2:  Now, we multiply the fraction's denominator on the right by the fraction's numerator on the left. We can represent 10/16 as 10/1 (since they are equivalent fractions).

 
Step 3: After the cross multiplication, always compare the LHS and RHS.
If they are equal, we can conclude that the fractions are equivalent. Here, in this 80 = 80, the fractions are equivalent.
 

The cross-multiplication formula is used for solving linear equations with two variables, as given below: 

(a)1x + (b)1y + (c)1 = 0
(a)2x + (b)2y + (c)2 = 0

The cross-multiplication formula we often apply is:

x(b1c1 - b2c2)  = y(c1a2 - c2a1)  = 1(b2a1 - b1a2) 

How to Derive Cross-Multiplication Formula?

In linear equations with two variables, we derive the cross-multiplication formula by eliminating one variable, often by making the coefficients of that variable equal.
For example:
Let’s consider two linear equations as
(a)1x + (b)1y + (c)1 = 0…(1)
(a)2x + (b)2y + (c)2 = 0…(2)
We solve these equations by making the coefficients of y equal in both equations:
We first multiply equation (1) by b2 and equation (2) by b1:
(b2)(a)1x + (b2)(b)1y + (b2)(c)1 = 0…(3)
(b1)(a)2x + (b1)(b)2y + (b1)(c)2 = 0…(4)
Then, subtract equation (4) from equation (3), 
(b2a1 – b1a2)x + (b2c1 – b1c2) = 0
Isolating x,
x = (b1c2 – b2c1) / (b2a1 – b1a2)…(a)
where (b2a1 – b1a2) ≠ 0
Similarly, solve (1) and (2) for y:
y = (c1a2– c2a1) / (b2a1 – b1a2)…(b)
where (b2a1 – b1a2) ≠ 0
We now combine (a) and (b),
x(b1c1 - b2c2)  = y(c1a2 - c2a1)  = 1(b2a1 - b1a2) 
Thus, we derived the required cross-multiplication formula.
 

Cross-multiplication is especially helpful in linear equations with two variables and is a quick way to find the solution. Solving linear equations with two variables, we apply the cross-multiplication method as given below:
Use the cross-multiplication formula:
x/(b1c2 – b2c1) = y/(c1a2– c2a1) = 1/(b2a1 –  b1a2)
Solve the linear equations:
3x + 4y = 8
2x + y = 5
Now, we convert the equations into standard form:
Since the general form of a linear equation is:
ax + by + c = 0
We rewrite the given equation as:
3x + 4y –8 = 0
2x + y – 5 = 0
When comparing the general form a1x + b1y = – c1 and a2x + b2y = –c2:
a1=3, b1​=4, c1​= −8
a2 = 2, b2=1, c2= −5
Now, apply the cross-multiplication formula:
x/(b1c2 – b2c1) = y/(c1a2– c2a1) = 1/(b2a1 –  b1a2)
Substituting the values:
x/(4) (– 5) – (1) (– 8) = y/(– 8) (2) – (– 5) (3) = 1/ (1) (3) – (4) (2)
Simplifying the fractions:
x/(–20 + 8) = y/(– 16 + 15) = 1/ (3 – 8)
x/(–12) = y/(– 1) = 1/ (– 5)
Isolating x:
Comparing the first and third fractions:
x/ –12 = 1/–5
x = (–12 × 1)/ –5 = 12/5
Isolating y:
Compare the second and third fractions:
y/ – 1 =1/ –5
y = –1 × 1/ –5 = 1/5
x = 12/ 5, y = 1/5
 

Cross-multiplication in equations helps students determine the value of unknown numbers easily. This technique can be applied to various real-life situations. Here are a few examples of its applications:

• Cross-multiplication is used to compare the prices of different items for choosing the cheaper option.

• This method helps adjust the quantities of ingredients required for a recipe without changing the proportions.

• Cross-multiplication is utilized in calculating the time required for a journey at a constant speed.

• We apply cross-multiplication to convert currencies accurately on the basis of their exchange rates.